Hi,

Could someone assist me in the right direction concerning the following problem:

Deduce the existence of a supremum from the principle of nested intervals.

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- Jan 5th 2011, 02:00 AMmusSupremum
Hi,

Could someone assist me in the right direction concerning the following problem:

**Deduce the existence of a supremum from the principle of nested intervals**. - Jan 5th 2011, 03:13 AMAlso sprach Zarathustra
I hope this will help...

Let be bounded sequence. Then exist and he the greater partial limit of the sequence.

In other words:

Proof:

Due to Bolzano-Weierstrass Theorem we may discuss on the set of partial limits, which is non-empty.

Given there are infinite that , but only a finite number of . Therefor there is infinite in neighborhood of , hence is partial limit.

Now, suppose that other partial limit then we will choose so that , but we know that there is only finite number of , therefor it impossible that is partial limit, thus is the greatest partial limit. - Jan 5th 2011, 04:12 PMmus
Could you kindly elaborate on the proof. Its not quite clear.

Thanks